Why would the sun burn out?
It runs out of fuel. The Sun is a low mass star, but there are TWO nuclear fusion processes going on in the Sun. The proton proton chain reaction fuses four protons into one helium nucleus with two protons and two neutrons. The carbon nitrogen cycle takes place in the inner core that requires higher temperatures and pressures. Carbon, nitrogen and oxygen are already present and are catalysts. They are not created by the process. They are recycled. The Sun will expand into red giant, but no one really knows how far it will expand. Then there will be a helium flash, not a carbon flash, when the Sun tries to start fusing hydrogen into helium in the layer around the core. Then TWO things will happen. The outer conductive and convective layers explode outward, creating a planetary nebula, and the core collapses until degeneracy pressure stops the collapse. The core becomes a white dwarf star. NO fusion is going on. The core cools off for billions of years and becomes a black dwarf star.
Running out of Hydrogen
The Sun is a giant nuclear reactor. It is making helium from hydrogen and emitting energy. When all the hydrogen is used up , then helium will react to form carbon (Red Giant stage)
The sun is a fusion reaction. If you take two hydrogen atoms and press them together really really hard, under high heat, they will 'fuse' together into a helium atom. And release a lot of energy in the process, which will make adjacent helium atoms fuse. That's what makes the sun hot and bright. Eventually all the hydrogen will be converted to helium, or at least enough that there can no longer be a 'chain reaction'. And then the sun will go dark. It will become just a cloud of helium. (Basically. This is a huge oversimplification, but good enough for our purposes.) When I was in college we were told this would happen in only a billion years. But more recently they've increased that to 5 billion years. I feel much better! As the sun burns out, it will expand bigger and bigger. It will envelop the earth and the earth will go Phhffft like a match head. Before it burns out.
Because it only has a finite amount of fuel. It has nothing to do with the lack of Sunspots and is certainly not the cause of cold weather in Eastern US, which may be due to global warming causing the polar vortex to break down. It will be billions of years before the Sun runs out of fuel. Before it does, it will become a red giant, and may engulf Earth.
The nuclear fusion in the sun will stop when it runs out of hydrogen. It will fuse helium and other elements for a while but then eventually become a white dwarf and slowly cool.
The sun is playing a balance between an expanding thermonuclear fireball and gravity which holds it in check to its current size. ---Should the mass decrease as the fuel is exhausted, the gravity cant hold it and it expands into a red giant, swallowing up everything up to the orbit of MARS. Should the nuclear reaction stop, or slow down it will collapse in on itself via gravity and become a white dwarf or even a black hole
The sun is consuming it's fuel - hydrogen - at a rate of about 4 tons per second. It's thought that the sun will continue burning in such a manner for about 4.5 billion more years; after that, it will run low on hydrogen, and shrink a little. The added pressure on the core will induce the helium building there to begin fusing into carbon; the 'carbon flash' will expand the sun to the point that it consumes Earth. Once the helium stops burning, all that will be left is the carbon/helium core of the sun - about the size of Earth. This is called a white dwarf... it'll cool over billions of years.
It would not, the sun is a massive nuclear fusion reactor. It can't just "burn out". It can appear less bright due to clouds in our atmosphere.
The same reason anything burns out. It runs out of fuel. Which of course it eventually will.
Running out of fuel?
People is too bad causing the planets to not rotate properly.